# Oct. 22, 2002
# Solutions to the wave equation u_tt=4u_xx using D'Alembert's solution.
# 
# 
> f:=x->piecewise(x<-Pi,0,x<Pi,sin(x),0);
> plot(f(x),x=-10..10);
# 
# The function u below solves the wave equation, and has initial
# velocity equal to zero.
# 
> u:=(x,t)->(f(x+2*t)+f(x-2*t))/2;
> plot([u(x,0),u(x,2),u(x,15)],x=-40..40,color=[red,green,blue]);
# 
# The function uu below solves the wave equation, and has initial
# position equal to zero.
# 
> uu:=(x,t)->(0.25)*Int(f(s),s=x-2*t..x+2*t);
> plot([uu(x,0),uu(x,2),uu(x,15),uu(x,0.25)],x=-40..40,color=[red,green,
> blue,black]);
# 
# The following lines were used to compute the Rayleigh quotient for the
# function h, with p(x) being k.
# 
> k:=x->piecewise(x<0,0,x<1/2,1,x<1,2,0);
> h:=(1-x)*x^2;
> Int(k(x)*diff(h,x)^2,x=0..1)/Int(h^2,x=0..1):evalf(%);
>